[Abridged.] We present multiwavelength observations of the black hole binary system, A0620-00. Using the Cosmic Origins Spectrograph on the Hubble Space Telescope, we have obtained the first FUV spectrum of A0620-00. The observed spectrum is flat in the FUV and very faint (with continuum fluxes ≃ 1e - 17 ergs/cm^2/s/A). We compiled the dereddened, broadband spectral energy distribution of A0620-00 and compared it to previous SEDs as well as theoretical models. The SEDs show that the source varies at all wavelengths for which we have multiple samples. Contrary to previous observations, the optical-UV spectrum does not continue to drop to shorter wavelengths, but instead shows a recovery and an increasingly blue spectrum in the FUV. We created an optical-UV spectrum of A0620-00 with the donor star contribution removed. The non-stellar spectrum peaks at ≃3000 {\deg}A. The peak can be fit with a T=10,000 K blackbody with a small emitting area, probably originating in the hot spot where the accretion stream impacts the outer disk. However, one or more components in addition to the blackbody are needed to fit the FUV upturn and the red optical fluxes in the optical-UV spectrum. By comparing the mass accretion rate determined from the hot spot luminosity to the mean accretion rate inferred from the outburst history, we find that the latter is an order of magnitude smaller than the former, indicating that ∼90% of the accreted mass must be lost from the system if the predictions of the disk instability model and the estimated interoutburst interval are correct. The mass accretion rate at the hot spot is 10^5 the accretion rate at the black hole inferred from the X-ray luminosity. To reconcile these requires that outflows carry away virtually all of the accreted mass, a very low rate of mass transfer from the outer cold disk into the inner hot region, and/or radiatively inefficient accretion. ; Comment: ApJ, accepted